Open Access
August 2019 Uniform behaviors of random polytopes under the Hausdorff metric
Victor-Emmanuel Brunel
Bernoulli 25(3): 1770-1793 (August 2019). DOI: 10.3150/18-BEJ1035


We study the Hausdorff distance between a random polytope, defined as the convex hull of i.i.d. random points, and the convex hull of the support of their distribution. As particular examples, we consider uniform distributions on convex bodies, densities that decay at a certain rate when approaching the boundary of a convex body, projections of uniform distributions on higher dimensional convex bodies and uniform distributions on the boundary of convex bodies. We essentially distinguish two types of convex bodies: those with a smooth boundary and polytopes. In the case of uniform distributions, we prove that, in some sense, the random polytope achieves its best statistical accuracy under the Hausdorff metric when the support has a smooth boundary and its worst statistical accuracy when the support is a polytope. This is somewhat surprising, since the exact opposite is true under the Nikodym metric. We prove rate optimality of most our results in a minimax sense. In the case of uniform distributions, we extend our results to a rescaled version of the Hausdorff metric. We also tackle the estimation of functionals of the support of a distribution such as its mean width and its diameter. Finally, we show that high dimensional random polytopes can be approximated with simple polyhedral representations that significantly decrease their computational complexity without affecting their statistical accuracy.


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Victor-Emmanuel Brunel. "Uniform behaviors of random polytopes under the Hausdorff metric." Bernoulli 25 (3) 1770 - 1793, August 2019.


Received: 1 September 2017; Revised: 1 March 2018; Published: August 2019
First available in Project Euclid: 12 June 2019

zbMATH: 07066239
MathSciNet: MR3961230
Digital Object Identifier: 10.3150/18-BEJ1035

Keywords: computational geometry , Convex bodies , Convex hull , deviation inequality , Hausdorff metric , high dimension , minimax estimation , random polytope

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 3 • August 2019
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